Dynamic response of various von-Ka´rma´n non-linear plate models and their 3-D counterparts

Dynamic von-Ka´rma´n plate models consist of three coupled non-linear, time-dependent partial differential equations. These equations have been recently solved numerically [Kirby, R., Yosibash, Z., 2004. Solution of von-Ka´rma´n dynamic non-linear plate equations using a pseudo-spectral method. Comp. Meth. Appl. Mech. Eng. 193 (6–8) 575–599 and Yosibash, Z., Kirby, R., Gottlieb, D., 2004. Pseudo-spectral methods for the solution of the von-Ka´rma´n dynamic non-linear plate system. J. Comp. Phys. 200, 432–461] by the Legendre-collocation method in space and the implicit Newmark-b scheme in time, where highly accurate approximations were realized. Due to their complexity, these equations are often reduced by discarding some of the terms associated with time derivatives which are multiplied by the plate thickness squared (being a small parameter). Because of the non-linearities in the system of equations we herein quantitatively investigate the influence of these a-priori assumption on the solution for different plate thicknesses. As shown, the dynamic solutions of the so called ‘‘simplified von-Ka´rma´n’’ system do not differ much from the complete von-Ka´rma´n system for thin plates, but may have differences of few percent for plates with thicknesses to length ratio of about 1/20. Nevertheless, when investigating the modeling errors, i.e. the difference between the various von-Ka´rma´n models and the fully three-dimensional non-linear elastic plate solution, one realizes that for relatively thin plates (thickness is 1/20 of other typical dimensions), this difference is much larger. This implies that the simplified von-Ka´rma´n plate model used frequently in the literature is as good as an approximation as the complete (and more complicated) model. As a side note, it is shown that the dynamic response of any of the von-Ka´rma´n plate models, is completely different compared to the linearized plate model of Kirchhoff–Love for deflections of an order of magnitude as the plate thickness.